Exceptional Splitting of Reductions of Abelian Surfaces with Real Multiplication

2018-07-16 10:00 - 2018-07-16 11:30 Room 78201, Jingchunyuan 78, BICMR Abstract: Zywina showed that after passing to a suitable field extension, every abelian surface $A$ with real multiplication over some number field has geometrically simple reduction modulo $\\frak{p}$ for a density one set of primes $\\frak{p}$. One may ask whether its complement, the density zero set of primes $\\frak{p}$ such that the reduction of $A$ modulo $\\frak{p}$ is not geometrically simple, is infinite. Such question is analogous to the study of exceptional mod $\\frak{p}$ isogeny between two elliptic curves in the recent work of Charles. In this talk, I will show that abelian surfaces over number fields with real multiplication have infinitely many non-geometrically-simple reductions. This is joint work with Ananth Shankar.\n